non-monotonicity of fertility in human capital accumulation and economic growth

Journal of Macroeconomics xxx (2013) xxx–xxx

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Journal of Macroeconomics

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Non-monotonicity of fertility in human capital accumulationand economic growth

0164-0704/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmacro.2013.06.006

⇑ Corresponding author. Address: University of Milan, Department of Economics, Management and Quantitative Methods (DEMM), via ConservI-20122 Milan, Italy. Tel.: +39 (0)2 50321 463; fax: +39 (0)2 50321 505.

E-mail address: [email protected] (A. Bucci).1 More recent examples of the view that population growth is detrimental to economic development include the neoclassical growth theory with e

technological progress (Solow, 1956), some empirical applications of this theory (notably, Mankiw et al., 1992), and Barro and Becker (1988, 1989). Theauthors consider an environment in which fertility is endogenous. Under the assumption that children are normal goods and parents are altruistic (parabout their children’s well-being), they conclude that a faster population growth, by implying a dilution of the capital endowment of each individupopulation, harms long-run growth in real per capita incomes.

2 See Kuznets (1960, 1967), Simon (1981), Boserup (1981), Kremer (1993), and Jones (2001), just to mention a few examples.

Please cite this article in press as: Boikos, S., et al. Non-monotonicity of fertility in human capital accumulation and economic gJournal of Macroeconomics (2013), http://dx.doi.org/10.1016/j.jmacro.2013.06.006

Spyridon Boikos a, Alberto Bucci a,⇑, Thanasis Stengos b

a Department of Economics, Management and Quantitaive Methods (DEMM), University of Milan, Italyb Department of Economics, University of Guelph, Ontario, Canada

a r t i c l e i n f o

Article history:Available online xxxx

JEL classification:O41J13J24

Keywords:FertilityPopulation growthEconomic growthHuman capital investmentHuman capital dilution

a b s t r a c t

This paper investigates the relationship between per capita human capital investment andthe fertility rate. In the first part of the article we analyze a theoretical model with endog-enous birth rate in which we do not make any assumption on how fertility directly affectsper capita human capital accumulation. The results obtained in this model are then com-pared with those of a more traditional setting where the birth rate is exogenous and inwhich the direct effect of this variable on per capita human capital investment is monoton-ically negative, a rather standard assumption within the available theoretical literature. Byusing non-parametric techniques, we document the presence of a strong non-monotonicityin the total effect that fertility plays on human capital accumulation, and hence on eco-nomic growth. The non-monotonic effect of fertility on human capital appears to holdempirically for OECD, as well as non-OECD countries.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The analysis of the impact of demographic change (population growth) on the growth rate of real per capita income rep-resents an old, but still unsettled topic of research. Malthus (1798) was among the first to recognize that a higher populationgrowth rate would ultimately in the very long-run have led to economic stagnation. According to his view, in a world inwhich economic resources are in fixed supply and technological progress is very slow or totally absent, the food-productionactivity would, sooner or later, have been overwhelmed by the pressures of a rapidly growing population. In this scenario,the available diet of each single individual in the population would have fallen below a given subsistence level, so leading to afall of the productivity growth rate as well.1 Unlike Malthus, proponents of the optimistic view2 emphasize, instead, the po-sitive effect that a larger population can exert on the rate of technological progress (an endogenous variable) and, thus, on eco-nomic growth: ‘‘. . .More people means more Isaac Newtons and therefore more ideas. More ideas, because of nonrivalry, mean moreper capita income. Therefore, population growth, combined with the increasing returns to scale associated with ideas delivers sus-

atorio 7,

xogenouslast two

ents careal in the

rowth.

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2 S. Boikos et al. / Journal of Macroeconomics xxx (2013) xxx–xxx

tained long-run growth’’ (Jones, 2003, p. 505).3 Besides the pessimistic and optimistic ones, there also exists another belief aboutthe long-run effects of population growth on economic growth: the neutralist one. The advocates of this view claim that pop-ulation growth has in general only little significant impact on economic growth and that such impact can be either positive, ornegative, or else wholly inexistent (Srinivasan, 1988; Bloom et al., 2003, p. 17).

The pessimistic prediction of Malthus (1798) has fortunately never become reality. Nonetheless, it is now well known(Bloom et al., 2003, Fig. 1.1, p. 13; United Nations, 2004) that, unlike the industrialized world, the less (and especially theleast) developed regions of the planet are rapidly and increasingly gaining shares of the world population. Since these re-gions are those that actually exhibit the highest fertility and the lowest literacy and economic growth rates, the followingquestion becomes of paramount importance in such a context: what is the effect that a further increase in the fertility rate(thus, in the population growth rate) may have on human capital accumulation and, through this channel, on sustained long-run economic growth?

The main objective of the present paper is to tackle this issue from a theoretical as well as an empirical perspective. Inorder to achieve this objective we briefly present the main results of a simple benchmark model in which individuals caninvest solely in human capital, the only input in the aggregate production function. We also assume that final output (aggre-gate GDP) can only be consumed, that the economy is closed (there is no international trade in goods and services and nointernational migration of people) and, finally, that there exists no governmental activity. Thus, in such an environmentthere is no room for physical capital investment. In this model the birth rate is exogenous, and moreover its direct effecton per-capita human capital investment (the so-called dilution effect) is taken as linear and monotonically negative. Thisis a rather standard assumption in the growth literature with human capital accumulation. The main prediction of the modelis that the total impact of the population growth rate (the birth rate) on the rate of economic growth is, in general, mono-tonically negative as well.4

Then, we extend the benchmark model by allowing for an endogenous birth rate, and by making no ex-ante specificassumption on the way this rate might affect directly per capita human capital investment. This modeling strategy is com-patible with a situation in which the total impact of population growth on economic growth is non-monotonic, i.e. a situa-tion where the growth-effect of higher fertility is, as a whole, different (in sign and in magnitude) across different groups ofcountries characterized by different birth rates. Kelley (1988, p. 1686) was among the first to admit that the relationshipbetween population and economic growth rates is indeed non-monotonic.5

In the last section of the paper we go to the data. By using semi-parametric and fully non-parametric methods, our empir-ical investigation finds strong evidence against the exogeneity of the birth rate, as well as against the monotonicity of thetotal impact that the birth rate has on the rate of per capita human capital investment and economic growth (these arethe two building-blocks on which the simple benchmark model is founded).6

Kalaitzidakis et al. (2001), using non-parametric techniques, were among the first to search empirically for a nonlineareffect of human capital accumulation on economic growth. In their paper, the authors split human capital by gender (malevs. female human capital) and by category (primary vs. tertiary education). They argue that the nonlinearities come espe-cially from the distinction of human capital by gender (as a result of the discrimination between males and females inthe labor market), and suggest that overall human capital investment has a negative impact on economic growth (due tothe fact that, particularly in countries with low levels of human capital, acquiring skills is generally considered as a rent-seeking economic activity). In our paper, contrary to Kalaitzidakis et al. (2001), we search for possible non-monotonic effectsof demographic factors (the birth rate) on economic growth. Starting from the idea that a change in the birth rate affectsagents’ decision of how much to invest in per-capita human capital, and realizing that a higher birth rate leads to a fasterdepletion of available resources (so that investing in education becomes more costly), we claim that the relation betweenbirth rate and per-capita human capital accumulation is non-monotonic. The presence of this non-monotonicity is, in turn,

3 In models with endogenous technological change population can affect economic growth in two distinct ways. In some papers (Romer, 1990; Aghion andHowitt, 1992; Grossman and Helpman, 1991) real per capita income growth depends positively on population size. This result (known as strong scale effect) isrejected on empirical grounds (Jones, 1995). In other papers, notably the semi-endogenous growth models (Kortum, 1997; Segerstrom, 1998), it is populationgrowth (as opposed to population size) to sustain economic growth in the very long-run (weak scale effect). Another branch of endogenous growth theory hasrecently analyzed the impact of population growth on economic growth in environments in which there is also, along with technological progress, humancapital accumulation. Such papers (Dalgaard and Kreiner, 2001; Strulik, 2005; Bucci, 2008) find that population growth is not necessary for long-run economicgrowth and that the effect of population growth on economic growth can be either non-positive (Dalgaard and Kreiner, 2001), or ambiguous (Strulik, 2005;Bucci, 2008). In all these papers, however, population (size and/or growth) is an exogenous variable.

4 At most, a rise of the growth rate of population may have no effect on economic growth. This occurs when agents are perfectly altruistic towards futuregenerations, a very special case.

5 ‘‘[. . .] In some countries population growth may on balance contribute to economic development; in many others, it will deter development; and in still others, thenet impact will be negligible’’ (Kelley, 1988, p. 1686).

6 The economic consequences of demographic change are at the heart of many empirical studies. While in some of them the focus is on the economy as awhole (for instance, Brander and Dowrick, 1994; Kelley, 1988; Kelley and Schmidt, 2003), in others country-based, household-level surveys are presented inorder to shed light on the link between family size/fertility and human capital investment/children’s performance at school (see, among others, Rosenzweig andSchultz, 1987 for Malaysia; Goux and Maurin, 2004 for France and Stafford, 1987 for the US). In comparison to Brander and Dowrick (1994), who analyze theimpact of the birth rate on physical capital investment, we study the possible non-monotonic effects of the birth rate on human capital accumulation andeconomic growth. Contrary to Kelley (1988) and Kelley and Schmidt (2003), we use data covering a larger time-span (1960–2000) and non-parametricmethods. Finally, unlike the household-level, country-based surveys (that in general predict a monotonically negative relationship between the two variablesbeing analyzed), our main empirical motivation in the present paper is to search for possible non-monotonicities in the relation between birth rate and per-capita skill investment.

Please cite this article in press as: Boikos, S., et al. Non-monotonicity of fertility in human capital accumulation and economic growth.Journal of Macroeconomics (2013), http://dx.doi.org/10.1016/j.jmacro.2013.06.006


S. Boikos et al. / Journal of Macroeconomics xxx (2013) xxx–xxx 3

at the heart of the non-monotonicity in the relationship between birth rate and economic growth as well, and suggests thatthe opportunity cost of having children is different across countries with different birth rates.

The article is organized as follows. In Section 2 we present the main assumptions and predictions of a simple (benchmark)model in which all the demographic components of population growth are taken as exogenous and summarized in a singleparameter, n. This benchmark model represents a useful theoretical starting point and offers itself for comparison with aricher and more general model (Section 3) where the birth rate is endogenous. In Section 4, we empirically estimate the ef-fect of the birth rate on human capital accumulation. According to this analysis, we first of all find strong evidence that thebirth rate is endogenous. Moreover, we also observe that the impact of the birth rate on human capital accumulation at theindividual level is non-monotonic, a result that is contrary to the monotonically negative relationship revealed by the simplebenchmark model. The last section summarizes and concludes.

2. The benchmark model: exogenous birth rate and monotonically negative fertility effect

Consider an economy in which households purchase consumption goods and choose how much to invest in human cap-ital. Each individual in the population offers inelastically one unit of labor-services per unit of time. Hence, population (N)coincides with the available number of workers and grows at an exogenous and constant rate c = n � d. Population growth(c) depends on three fundamental variables: fertility (the birth rate, n), mortality (the death rate, d), and migration. We ne-glect migration (the economy is closed to international trade in goods and services and to international mobility of people),we abstract from any fertility decision (whereby rational agents choose the number of their descendants by weighing costsand benefits of rearing children), and consider the mortality rate as exogenous. Therefore, we take the growth rate of pop-ulation as given. Moreover, since in the paper we focus on the birth rate as the fundamental variable affecting agents’ invest-ment in education (in the empirical section of this work we estimate the total effect that a change in the birth rate has onper-capita human capital investment7), we simplify further the analysis by setting d = 0.8 Hence, in the remainder of the paperthe population growth rate equals the birth rate, Nt

�=Nt � c ¼ n.9

Following Barro and Sala-i-Martin (2004, Chap. 5, p. 240), the total stock of human capital existing at time t (Ht � Ntht)changes not only because population size can vary, but also because the average quality of each worker (or per capita humancapital, ht � Ht/Nt) may increase over time.

Consumption goods (or final output) are produced competitively by using solely human capital (HY) as an input. Theaggregate technology for the production of final output is:

7 Theincrease

8 ForEhrlich

9 In t(fertility

10 ‘‘. . .

exhibits

PleaseJourna

Yt ¼ AHYt ; HYt ¼ utHt ð1Þ

with Yt denoting the GDP of the economy at time t, A > 0 representing a positive productivity parameter (total factor produc-tivity), and ut being the share of the total stock of human capital devoted to the production of goods at each t P 0.

There is no physical capital investment and final output (the numeraire in this economy) can only be consumed. Theaggregate production function (Eq. (1)) is linear in the amount of human capital devoted to goods manufacturing. However,unlike ‘‘AK-type’’ growth models, individuals choose endogenously how to allocate the existing stock of (human) capital be-tween production of consumption goods (ut) and production of new human capital (1 � ut). Under our assumptions, theeconomy-wide budget-constraint reads as:

Yt ¼ AHYt ¼ AutHt ¼ AutNtht ¼ Ct; ð2Þ

where C is aggregate consumption.Concerning human capital accumulation, we follow Uzawa (1965) and Lucas (1988) and assume that the law of motion of

human capital at the economy-wide level is:

Ht

�¼ rð1� utÞHt; r > 0; ð3Þ

where r is a positive parameter representing the productivity of human capital in the production of new human capital.10

For the sake of simplicity we assume that human capital is not subject to any form of material obsolescence. Given Ht

�and the

definition of per capita human capital ht, the law of motion of per capita human capital is:

h�

t ¼H�

t

Nt� nht ¼ ½rð1� utÞ � n�ht:

birth rate is the main determinant of individuals’ human capital decisions according to some empirical surveys in which it is clearly shown that anin the family size leads to a reduction in children’s participation to education (Radyakin, 2007; Booth, 2005).

a comprehensive analysis of the effects of declining mortality (rising longevity) on investment in education and economic growth see, among others,and Lui (1991), Zhang et al. (2005), and Kalemli-Ozcan (2002).he absence of immigration and mortality, there is a one-to-one correspondence between the population growth rate and the number of children).

Uzawa (1965) worked out a model very similar to this one. The striking feature of his solution, and the feature that recommends his formulation to us, is that itsustained per-capita income growth from endogenous human capital accumulation alone: no external ’engine of growth’ is required’’ (Lucas, 1988, p. 19).

cite this article in press as: Boikos, S., et al. Non-monotonicity of fertility in human capital accumulation and economic growth.l of Macroeconomics (2013), http://dx.doi.org/10.1016/j.jmacro.2013.06.006



With n > 0, the term�nht represents the cost (in terms of per capita human capital investment) of upgrading the degree ofeducation of the newborns (who are uneducated) to the average level of education of the existing population (linear andmonotonically negative dilution in human capital accumulation at the individual level).

With a Constant Intertemporal Elasticity of Substitution (CIES) instantaneous utility function, the objective of the family-head is to maximize under constraint the household’s intertemporal utility deriving from per capita consumption:

11 Thrassumpc remai

12 Alg13 Acc

v e [0; 1

PleaseJourna

Maxfct ;ut ;htgþ1t¼0

U �Z þ1

0

c1�ht

1� h

� �Nm

t e�qtdt; q > 0; m 2 ½0; 1�; h > 0 ð4Þ

s:t: : h�

t ¼ ½rð1� utÞ � n�ht; r > 0; n P 0; ut 2 ½0; 1�; 8t P 0 ð5Þ

along with the transversality condition:

limt!1

khtht ¼ 0; ð6Þ

and the initial condition11:

hð0Þ > 0: ð7Þ

The household chooses the amount of per capita consumption and the share of human capital to be devoted to productionactivity (respectively ct � Ct/Nt and ut). Eq. (4) is the household’s intertemporal utility function and Eq. (5) represents the percapita human capital accumulation function. We denoted by 1/h the constant intertemporal elasticity of substitution in con-sumption. Following many existing examples in the literature (for example, Razin and Sadka, 1995, Chap. 13, footnote 1), wemake a formal distinction between the pure rate of time preference (q) and the degree of agents’ altruism towards futuregenerations, m. The limiting case of m = 1 defines the situation of perfect altruism (the family-head maximizes the discountedvalue of total utility, i.e. per capita utility multiplied by the aggregate family size), whereas the opposite limiting case of m = 0defines the minimal degree of altruism (the family-head maximizes the discounted value of per capita utility). Clearly,m e (0; 1) describes an intermediate situation between the previous two. Since u is a fraction, it must belong to the interval[0; 1]. In Eq. (5) the exogenous birth rate (n) is set at a value that can be either positive or, at most, equal to zero. By solvingthe benchmark model, in which

h�

t

ht� gh ¼ rð1� utÞ þ f ðnÞ; f ðnÞ � �n;

it is possible to show that along the balanced growth path (BGP) equilibrium12:

u ¼ rðh� 1Þ þ q� ðhþ v � 1Þnrh

� uðnÞ; gc ¼ gy ¼ gh ¼ðr� qÞ � ð1� vÞn

h� gðnÞ;

where gZ represents the growth rate of variable Z and yt � Yt/Nt is the level of per capita real income. Hence, exogenous pop-ulation growth (n) may overall have either a negative or at most no effect on economic growth (g). Intuitively, the total im-pact of population growth on real per capita income growth can de decomposed into two distinct effects:1. The first is represented by what we may label (indirect) accumulation effect. This effect is formally represented by:� �

@

@n½rð1� uðnÞÞ� ¼ @

@nðr� qÞ þ ðhþ v � 1Þn

h¼ 1

hðhþ v � 1Þ:

The (indirect) accumulation effect is monotonically positive, for any h + v > 1.13

2. The second is, instead, represented by the (direct) dilution effect. This effect is given by:

@

@n½f ðnÞ� ¼ @

@nð�nÞ ¼ �1 < 0:

The (direct) dilution effect is monotonically negative, for any value of the parameters.Given the previous two effects, the (total) fertility effect that a change in n may exert on real per-capita income growth (g)

is therefore equal to:

@gðnÞ@n|fflffl{zfflffl}

ðtotalÞfertility effect

¼ @

@nrð1� uðnÞÞ þ f ðnÞ|{z}

��n

24

35 ¼ @

@n½rð1� uðnÞÞ�|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðindirectÞaccumulation effect

þ @

@nð�nÞ|fflfflfflffl{zfflfflfflffl}

ðdirectÞdilution effect

¼ v � 1h6 0:

oughout the paper, we normalize population size (and, therefore, the number of workers) at time zero to one, i.e. we set N(0) � 1. Under thistion, the objective function in Eq. (4) can also be written as: U �

Rþ10

c1�ht

1�h

� �e�ðq�mnÞtdt. In this case q > mn ensures that U is bounded away from infinity if

ns constant over time.ebraic derivation of the BGP solution of this model is simple and available from the authors upon request.ording to empirical studies, we now know that h is fairly large, and in some cases (e.g., Hall, 1988) it is definitely larger than one. Therefore, for any], assuming h + v > 1 seems in principle realistic.




We see that in the benchmark model, as far as h + v > 1, the monotonically negative ‘dilution’ effect always prevails over(or, at most, is equal to) the monotonically positive ‘accumulation’ effect, so that the (total) ‘fertility’ effect is also in generalmonotonically negative: only when v = 1, this effect equals zero (in this case the dilution and accumulation effects would havethe same magnitude, but would be opposite in sign, therefore they would exactly cancel out each other).

3. An extension: endogenous birth rate and non-monotonic fertility effect

In this section we extend the previous model along two directions: (i) We endogenize the birth rate (n) and, more impor-tantly, (ii) We do not make any specific assumption about the form that function f(n) can undertake in the law of motion ofper capita human capital. This is the most important difference with respect to the benchmark model: while in that modelthe direct dilution effect was a priori assumed to be linearly monotonically negative, that is f(n) = �n, and found to prevail, forany h + v > 1 and v e [0; 1), on the monotonically positive accumulation effect (so that the resulting fertility effect was alsomonotonically negative), in the model of this section we do not make any hypothesis on the sign of the direct effect that achange in n may yield on per capita human capital investment. As a consequence, the (total) fertility effect can in principlebehave in a totally non-monotonic way as a consequence of a change in the birth rate. As far as we know, this is the firstattempt in the literature at analyzing and characterizing a similar model.14

To be more concrete, we postulate that:

14 Thewith a sthe reprof any ocapital

15 Sincjustificauneduc(ceterisintensivpoliciesparticip

16 It isderivati

17 Theutility f

18 See

PleaseJourna

h�

t ¼ ½rð1� utÞ þ f ðntÞ�ht ; ð8Þ

where the term f(n) is left unspecified. This means that any possibility concerning the sign of the dilution effect – namely, thesign of @

@n ½f ðnÞ� – is now open, including non-monotonicity.15 The only assumption we introduce at this stage is f 00ðnÞ 6 0,which guarantees that the intertemporal agents’ problem has a maximum.16 As whole, the fact that we use f 00ðnÞ 6 0 whilesimultaneously not imposing any ‘structure’ on f(n) entails that the dilution effect can be either linear or nonlinear (secondderivative equal to, or lower than zero, respectively), and is potentially non-monotonic (first derivative neither always positive,nor always negative for any admissible value of n).

Apart from these, the other assumptions on the economy remain the same as in the benchmark model. In particular, theaggregate production function and the aggregate budget constraint (Eqs. (1) and (2), respectively) are unchanged, agents de-vote their own time-endowment in part (ut) to production activities and in part (1 � ut) to produce new human capital, theeconomy is closed, there is no migration of people across countries, and the exogenous death rate is normalized to zero(d = 0), so that the rate of population growth keeps on corresponding to the birth rate.

Because it is endogenous, we follow Palivos and Yip (1993) in considering the birth rate, along with per capita consump-tion, as an argument of the individual instantaneous utility function. However, we depart from Palivos and Yip (1993) in thatwe analyze the predictions of a model with endogenous allocation of human capital between production and education sec-tors. More specifically, the instantaneous utility function of each individual in the economy is postulated to be:

uðct ; ntÞ �ðcb

t n1�bt Þ1�h

1� h; b 2 ð0; 1�; h 2 ð0; 1Þ; ð9Þ

where b and (1 � b) determine, respectively, the weights by which c and n enter the individual utility function. As it is wellknown, the utility function (9) exhibits a constant elasticity of marginal utility for both c and n.17 The hypothesis b e (0; 1]suggests that per-capita consumption is a fundamental argument of each individual’s instantaneous utility. As in Palivos andYip (1993), h must be different from one for an equilibrium value of n to exist (see next Eq. (14), and Footnote 16). For positivevalues of ct and nt, and with b e (0, 1), the assumption 0 < h < 1 guarantees also that u(�) remains always strictly positive.18 Theevolution of the population size (N) over time is governed by:

N�

t ¼ ntNt ; nt P 0: ð10Þ

original Lucas’ (1988, Eq. 13, p. 19) formulation does not include any dilution effect at all. Lucas’ assumption (newborns enter the workforce endowedkill-level proportional to the level already attained by older members of the family, so population growth per se does not reduce the current skill level ofesentative worker) is based on the ‘social nature’ of human capital accumulation, which has no counterpart in the accumulation of physical capital andther form of tangible assets. On the other hand, the assumption of a monotonically negative dilution effect of population growth on per capita human

investment can be found in Strulik (2005, p. 135, Eq. 24) and Bucci (2008, p. 1134, Eq. (120)), just to mention two examples.e in the present context the dilution effect may be non-monotonic, the occurrence of an eventually positive sign for this effect deserves some economiction. Indeed, in the literature the dilution effect is generally considered negative because (the argument goes) newborns enter the world totallyated, and hence reduce (dilute) the existing stock of per capita human capital. Put differently, the higher the number of children in a family, and the lessparibus) that family’s available income that can be devoted to the education of each child. However, it is also true that the cost of raising children is moree in time than in material goods (Robinson, 1987). In addition, mostly in advanced countries, there now exist important children’s governmentalespecially targeted to support the income of those families having a large number of kids and that, as a consequence, aim at increasing parents’ laboration (see Fondazione G. Brodolini, 2005).possible to show that the assumptions h e (0; 1) and f 00ðnÞ 6 0 are sufficient to assure the concavity of the Hamiltonian function. Detailed algebraic

on of this result is obtainable from the authors upon request.elasticity of marginal utility of consumption equals 1 � b(1 � h), whereas n has an elasticity of marginal utility equal to b + h(1 � b). When b = 1 the

unction becomes uðct ; ntÞ � c1�ht

1�h ¼ uðctÞ and 1/h > 0 is the usual elasticity of substitution in consumption.Growiec (2006, pp. 10, 16 and 17), and Jones (2001, p. 4, Eq. (1)).




The intertemporal problem faced by the representative family-head is now:

19 In Acoming

20 In oeffectsdilution

PleaseJourna

Maxfct ;ut ;nt ;ht ;Ntgþ1t¼0

U �Z þ1

0

ðcbt n1�b

t Þ1�h

1� hNm

t e�qtdt; q > vn P 0; m 2 ½0; 1�; b 2 ð0; 1�; h 2 ð0; 1Þ ð11Þ

s:t: : N�

t ¼ ntNt; nt P 0 8t P 0 ð10Þ

h�

t ¼ ½rð1� utÞ þ f ðntÞ�ht; r > 0; ut 2 ½0; 1�; 8t P 0 ð8Þ

along with the two transversality conditions:

limt!1

khtht ¼ 0; limt!1

kNtNt ¼ 0;

and the initial conditions:

hð0Þ > 0; Nð0Þ ¼ 1 > 0:

3.1. BGP analysis

We employ the following definition of BGP equilibrium.

Definition (BGP Equilibrium). A BGP equilibrium is an equilibrium path along which: (i) All variables depending on timegrow at constant (possibly positive) exponential rates; (ii) The allocation of human capital between production ofconsumption goods and production of new human capital is constant (ut = u, "t P 0).

Proposition 1. Along the BGP equilibrium:

u ¼ bðh� 1Þ½rþ f ðnÞ� þ q� vnr½bðh� 1Þ þ 1� ð12Þ

gc ¼ gy ¼ gh � g ¼ ðr� qÞ þ vnþ f ðnÞ½bðh� 1Þ þ 1� ð13Þ

f 0ðnÞ ¼ vbðh� 1Þ �

1� bbn

� �rbðh� 1Þ þ bðh� 1Þf ðnÞ þ q� vn

bðh� 1Þ þ 1

� �: ð14Þ

Detailed derivation of the results stated in Proposition 1 can be found in an Appendix available from the authors uponrequest.

Eqs. (12) and (13) provide respectively the fraction of human capital (u) devoted by each individual in the population tonon-educational activities (namely, production of consumption goods), and the common growth rate (g) attained by theeconomy in per-capita terms over the very long-run (BGP equilibrium), whereas Eq. (14) would give, by knowing the exactform of f(n), the BGP equilibrium values of the endogenous birth rate(s), n P 0. Note that in this model (as in the benchmarkcase) economic growth is driven by human capital accumulation (that is, h

�t=ht � gh ¼ gy � g).

It is apparent from Eq. (14) that the current model may display multiplicity of equilibria. In particular, if f0(n) were a qua-dratic function of n (as it is going to be the case empirically, see the next section and in particular Table 2), Eq. (14) would besatisfied (for given b, m, h, r and q) for three different values of n.19 However, it is not the objective of this paper to build atheory of the role of multiplicity in economic growth, since we are mainly focused here to highlight the possible non-monotoniceffects of the birth rate on real per capita income growth through per capita human capital investment. In this respect, mim-icking the analysis already done in the benchmark model with exogenous fertility, we can decompose the total effect that achange in n has on per capita human capital accumulation and economic growth into two separate effects.20 However, beforedoing that, from Eqs. (12) and (13), we clearly see that: u = u(n), and g = g(n). Thus, according to these two equations, every

ppendix A (Table A.1) of Boikos et al. (2012) it is shown that, under specific combinations of the parameter values, multiple (real and positive) roots of nfrom Eq. (14) do exist.rder to allow for a direct comparison of the results of this section with those of the previous one, we continue here to identify the direct and indirect

of a variation in n on per capita human capital investment and economic growth as, respectively, the dilution and accumulation effects. Concerning theeffect, see also the previous Footnote 15.




endogenous variable depends in this model on n, which in turn is a function of exogenous parameters. In particular, from Eq.(14), it is immediate to conclude that:

21 In tsign of

22 Becn = n(e)

PleaseJourna

n ¼ nðm; b; h;r;qÞ:

After defining by e the vector of these parameters, i.e.:

e � ðm;b; h;r;qÞ

we can recast in a more compact way n as:

n ¼ nðeÞ:

Finally, since also in the model of this section gc = gy = gh � g = r(1 � u) + f(n), in the end we can write

g ¼ r½1� uðnðeÞÞ� þ f ðnðeÞÞ � gðnðeÞÞ;

and compute (through the so-called chain-rule) the following derivative:

Given all this, we can now describe more precisely how the accumulation and dilution effects do work in the model of thissection. Suppose there is a change in e (namely, a change in one of the exogenous parameters capable of affecting the endog-enous value of n along the BGP, i.e. a change either in m, or in b, or in h, or in r, or else in q). This change in e:

1. Affects directly n, and therefore [f(n)]. This is the (direct) dilution effect, represented formally by the term @@n ½f ðnÞ� in paren-

theses, for given @n@e;

2. However, the changed n also affects u(n), and hence r[1 � u(n)]. This is the (indirect) accumulation effect, represented for-mally by @

@n ½rð1� uðnÞÞ� in parentheses, for given @n@e. Unlike the dilution effect, the accumulation effect is indirect because

the initial change in e affects u(n) only indirectly, through n.

As already said, in this section of the paper we let the (direct) dilution effect, and therefore the (total) fertility effect befree, in other words we do not impose any restriction on the way f ðnðeÞÞ and hence r[1 � u(n(e))] + f(n(e)) may react to achange in n due to exogenous reasons, that is an exogenous variation of e � (m,b,h,r,q). This means that, unlike the modelof the previous section, the one of this section is in principle compatible with a situation in which the (total) fertility effectthat a variation in n has on per capita human capital accumulation and economic growth is non-monotonic.21 At this stage, itis worth mentioning that the effects of changes in the model’s parameters (m,b,h,r,q) on the balanced growth rate of the econ-omy crucially depend on the sign of the dilution effect, @

@n ½f ðnÞ� – detailed derivation of this result can be obtained from theauthors upon request.

Proposition 2 ensures that the endogenous variables of the model undertake economically meaningful values along theBGP.22

Proposition 2. Assume r > q � vn > 0. The following inequality:

�ðr� qþ mnÞ < f ðnÞ < q� vn� rbð1� hÞbð1� hÞ ð15Þ

ensures that along the BGP equilibrium, g > 0 and u e (0;1) hold simultaneously.

he monotonic case o[f(n)]/on and o[r(1 � u(n))]/on would be either always positive or always negative, for any n. In the non-monotonic case, instead, thethese two derivates could be different depending on the specific range of n observed.ause we do not make any assumption on the specific functional form undertaken by f(n), we cannot compute in closed form the possible BGP value(s) of, e � (m,b,h,r,q). Hence, inequality (15) provides ultimately restrictions on the relations among the parameters of the model.



Table 1Semi/non-parametric regression results.

Variables OLS SP NP(A) (B) (C)

Constant 1.1521*** 0.6132*** –(0.0176) (0.0164)

doecd 0.0528*** 0.0883*** –(0.0115) (0.0168)

la 0.1709*** 0.1547*** –(0.0136) (0.0135)

Af 0.0153 0.0355*** –(0.0197) (0.0162)

d1965 0.0044 0.0078 –(0.0221) (0.0216)

d1970 0.0250 0.0281 –(0.0211) (0.0214)

d1975 0.0226 0.0219 –(0.0194) (0.0209)

d1980 0.0280 0.0252 –(0.0185) (0.0204)

d1985 0.0274 0.0268 –(0.0176) (0.0268)

d1990 0.0317 0.0318 –(0.0175) (0.0318)

d1995 0.0181 0.0168 –(0.0172) (0.0168)

cbr �16.3436*** – –(0.5328)

N 792 792 792R2/R2 adj. 72.37/71.98 74.2/73.7 77.66/–F-test 185.8***

Heterosked.a 158.96***

P(Specific.)b 2.22e�16*** 2.22e�16***

NP(Test)c 0.012531*

⁄⁄⁄ Denotes the 1% significance levels.⁄⁄ Denotes the 5% significance levels.⁄ Denotes the 10% significance levels.

a Heterosked. is the heteroskedasticity LM-test by Breusch and Pagan (1979). Robust standard errors are in parentheses.b The P(specific.) shows the p-values if the null of the parametric linear (linear-OLS) is correctly specified in comparison to a fully non-parametric (NP)

model, using the Hsiao et al. (2007) test for continuous and discrete data models after 399 Bootstrap replications. The P(specific.) at column B checks if theparametric model (linear-OLS) is well specified when compared to the semi-parametric (SP) model.

c The NP(Test) is a significance test for the explanatory variable (cbr) in the locally linear non-parametric specification. It is like the t-test in theparametric regression framework and is based on Racine (1997). We drop the time dummy for 2000.


Proof. Immediate from Eqs. (12) and (13). h

Proposition 3 draws attention to the links between the results of the model of Section 2 and those of the model discussedin the present section of the paper.

Proposition 3. For the BGP solution of the benchmark model (Section 2) with exogenous birth rate and f(n) = �n to be aspecific case of the BGP solution of the model (this section) with endogenous birth rate and a more general (unspecified) f(n),the knife-edge condition m = (1 � h) needs to be met, along with b = 1.

Proof. Simple algebraic inspection reveals that when b = 1 and f(n) = �n, Eqs. (12) and (13) deliver, respectively:u ¼ rðh�1Þþq�ðhþv�1Þn

rh , and gc ¼ gy ¼ gh ¼ ðr�qÞ�ð1�vÞnh . This is the BGP solution of the benchmark model with exogenous fertility.

Moreover, with b = 1, f(n) = �n, m = (1 � h), and n being taken as an exogenous constant, Eq. (14) – which in principle wouldallow us to determine the value of n as a function of the model’s parameters in a setting where fertility is endogenous – istrivially satisfied. A more formal and detailed proof of the Proposition, involving the analysis of the first order conditionstaken on the Hamiltonian function of the problem with endogenous birth rate, is available from the authors uponrequest. h

We can now move to empirical evidence.



Table 2Regression results by using polynomial terms for the birth rate (cbr).

Variables OLS OLS OLS OLS(A) (B) (C) (D)

Constant 1.1521*** 0.6958*** 1.1926*** 0.6108***

(0.0176) (0.0699) (0.0205) (0.1305)doecd 0.0528*** 0.0883*** �0.0630 0.1007

(0.0115) (0.0115) (0.02428) (0.1563)la 0.1709*** 0.1551*** 0.1626*** 0.1546***

(0.0136) (0.0140) (0.0140) (0.0143)af 0.0153 0.0301 0.0227*** 0.0312

(0.0197) (0.0195) (0.0198) (0.0195)d1965 0.0044 �0.0023 0.0022 �0.0018

(0.0221) (0.0219) (0.0219) (0.0222)d1970 0.0250 0.0180 0.0236 0.0194

(0.0211) (0.0209) (0.0210) (0.0211)d1975 0.0226 0.0139 0.0222 0.0147

(0.0194) (0.0193) (0.0193) (0.0195)d1980 0.0280 0.0199 0.0280 0.0237

(0.0185) (0.0182) (0.1844) (0.0172)d1985 0.0274 0.0234 0.0290 0.0237

(0.0176) (0.0171) (0.0175) (0.0172)d1990 0.0317 0.0300 0.0335 0.0307

(0.0175) (0.0170) (0.0173) (0.0171)d1995 0.0181 0.0165 0.0195 0.0163

(0.0172) (0.0166) (0.0169) (0.0166)cbr �16.3436*** 26.35** �17.4017*** 34.04*

(7.670) (13.3694)(0.5328) (0.6085)(cbr)2 �1143*** �1355**

(265.99) (419.65)(cbr)3 9224** 11050**

(2868.025) (4147.205)oecd(cbr) 5.1988*** 4.665

(1.040) (18.14)oecd(cbr)2 �403.77

(674.02)oecd(cbr)3 �6856

(7950)N 792 792 792 792R2/R2 adj. 72.37/71.98 73.87/73.43 72.84/72.42 73.91/73.37F-test 185.8*** 169.2*** 174.1*** 137.2***

F-Joint 158.96*** 22.269*** 13.371*** 0.4312Heterosked.a 133.24*** 183.37*** 167.11*** 185.55***

*** Denotes the 1% significance levels.** Denotes the 5% significance levels.* Denotes the 10% significance levels.

a Heterosked. is the heteroskedasticity LM-test by Breusch and Pagan (1979). Robust standard errors are in parentheses. We drop the time dummy for2000.


4. From theory to evidence: empirical methodology and data description

By focusing on the effects of the birth rate n on human capital accumulation,23 our main interest in this section is to esti-mate Eq. (8):

23 Evelife expthe crudcapita h

PleaseJourna

h�

t=ht ¼ rð1� utÞ þ f ðntÞ:

Since there are no available data for u (which is a function of n according to Eq. (12)), the previous equation and the anal-ysis of the preceding section imply:

n though in the theory presented above we set d = 0, in our empirical analysis we include the death rate as a control variable. More specifically, we useectancy as a proxy for the death rate. Alternatively, and for the sake of robustness, we also employed the crude death rate. Contrary to life expectancy,e death rate displays negative sign and is statistically significant. In both cases, however, we do find a non-monotonic effect of the birth rate on per-uman capital accumulation. The results with the death rate as a control variable are available upon request.




Hence, the equation we ultimately estimate is of the form

24 Thebusines

25 Whcapitalyears o

26 WetheoretSecondsecondaprimary

27 The28 We

(Brande29 Hum

variableused en

PleaseJourna

h�

t=ht ¼ /ðntÞ:

This equation captures at the same time the combined impact of the accumulation and dilution effects. We use an esti-mation method more flexible than OLS in order not only to check whether the relationship written above is mis-specified,but also to obtain a consistent estimate of the function that links the birth rate and human capital investment at the indi-vidual level. In addition, we also examine the possible endogeneity of n. However, before going into the details of theeconometric techniques and the specification tests used, we start by describing briefly the variables employed and thesources of our data.

Our sample consists of a panel of ninety-nine countries (both OECD and non-OECD) for the period 1960–2000. The obser-vations are averages over a 5 years-interval.24 Hence, we have 9 time periods. For human capital accumulation (which is ourdependent variable) we use enrollment rates for population aged between 15 and 65 years.25 The data about this variable comefrom the Barro and Lee (2000) dataset. It is well-known that the Barro–Lee dataset for human capital has an important limita-tion in the fact that it does not explicitly take into account differences in schooling-quality across countries. For this reason weuse group dummy variables for controlling for any potential difference between groups of countries. Despite its limitations, theBarro and Lee (2000) dataset has been extensively used in recent years and as such allows us to make direct comparisons withother empirical studies that explore the role of human capital in economic growth.26

According to de la Fuente and Domenech (2000), the stock measure of human capital (total mean years of schooling) suf-fers from serious measurement error problems, something that would be exacerbated if we were to obtain growth rates fromdifferencing the stock series. Hence, instead of measuring human capital accumulation in growth rates as it is the case forper-capita income, we prefer to use enrollment rates. Human capital accumulation for country i at time t is denoted byHUMi,t. The data for the crude birth rate (cbr) come from the UN (2008) dataset. The crude birth rate is measured as the num-ber of births over 1000 people.27 This is the variable that we estimate by using both OLS and semi/non-parametric methods.28

All the other variables enter linearly in our econometric specification.We use time dummies (dt) in order to capture any time specific effects and group dummies (dOECDi) to capture any dif-

ference between OECD and non-OECD countries. Following Kalaitzidakis et al. (2001), we also employ regional dummies(dRegioni) – such as dummies for Latin America (la) and Sub-Saharan Africa (af) countries – to control for possible hetero-geneity in our sample.

Finally, we use a vector Xi,t = (lifexp,hum60, infmort) of control variables that are used at a later stage in order to check forthe robustness of our results. These variables are life expectancy (lifexp), human capital at 1960 (hum60),29 and infant mor-tality (infmort), respectively. The data concerning the two demographic variables (life expectancy and infant mortality), likethose on the birth rate, come from the UN (2008) dataset. Life expectancy (lifexp) is defined as ‘‘the average number of yearsof life expected by a hypothetical cohort of individuals who would be subject during all their lives to the mortality rates of a givenperiod’’ (United Nations, 2008). This variable is expected to have a positive impact on human capital accumulation (intuitively,when expecting a higher chance of surviving, rational individuals invest more in education to enhance earnings for consumptionover a longer life. Moreover, the longer life expectancy, the longer the time-horizon within which the costly investment in hu-man capital can be re-paid). Infant mortality (infmort) is the probability of dying between birth and age one. It is expressed asthe number of deaths per 1000 births and it is expected to have a negative impact on human capital accumulation, since a rise ofinfant mortality would imply that the quantity-effect (having more children) dominates the quality-effect (having lower fertility,but more education per child). The average years of schooling at 1960 (hum60) is a stock measure of human capital and is usedin order to capture the difference in initial conditions across countries. Since this variable reveals the preference of a countrytowards human capital accumulation, it is expected to have a positive impact on our dependent variable.

The equation we estimate by OLS is the following:

HUMi;t ¼ b0 þ b1ðdOECDiÞ þ b2ðdRegioniÞ þ b3dt þ b4ni;t þ ei;t ;

use of averages over periods of 5 years, as opposed to the use of annual data, has been proposed in the literature as a strategy for reducing possibles cycle effects.en working with fertility it is better to use human capital data for population which is above 15 years old. We have tried another dataset for humanfrom the Barro and Lee (2000) data base for population above 25 years and the nonlinearities are also present there. The results for population over 25ld are available upon request.

use total human capital, which is the sum of primary, secondary and higher education. This is done for three major reasons. The first is mainlyical in nature and relies on the fact that in the models of the previous sections there is no distinction of human capital by category of education.ly, non-OECD countries exhibit very small participation rates at higher levels of education. Finally, because in every country participation at primary andry level of education is a necessary requirement in order for people to proceed into higher levels of schooling, we believe it is preferable to include alsoand secondary education in our measure of human capital.crude birth rate is expressed in percentage terms for comparability with the enrollment rates.have also used the adjusted birth rate proposed by Brander and Dowrick (1994):‘‘. . .Adjusted births are simply crude or gross births net of infant mortality’’r and Dowrick, 1994, p. 5). The results do not change and are available upon request.

an capital at 1960 (hum60) is the stock of human capital in the year 1960. It is measured as the average years of schooling in that year. We use this, and not the enrollment rate at 1960, in order to reduce any possible bias due to the correlation with the dependent variable. However, we have alsorollment rates at 1960, but results are not affected at all by this change.



Table 3Regression results using polynomial terms for the birth rate (cbr) and control variables.

Variables OLS OLS OLS OLS(A) (B) (C) (D)

Constant 0.33319** 0.0449 0.3824*** 0.0161(0.1454) (0.1504) (0.1413) (0.1754)

doecd 0.0038 0.0205 �0.0816*** 0.1449(0.0097) (0.0094) (0.0205) (0.1339)

la 0.1260*** 0.1158*** 0.1204*** 0.0113***

(0.0093) (0.0094) (0.0094) (0.1138)af 0.0610*** 0.0704*** 0.0643*** 0.0709***

(0.0119) (0.0114) (0.0118) (0.0115)d1965 �0.0180 �0.0298* �0.0233 �0.0274

(0.0164) (0.0165) (0.0163) (0.0162)d1970 �0.0132 �0.0231* �0.0172 �0.0204

(0.0162) (0.0161) (0.0160) (0.0162)d1975 �0.0154 �0.0246 �0.0179 �0.0218

(0.0155) (0.0154) (0.0153) (0.0155)d1980 �0.0098 �0.0170 �0.0113 �0.0149

(0.0146) (0.0141) (0.0144) (0.0144)d1985 �0.0110 �0.0147 �0.0108 �0.0131

(0.0139) (0.0136) (0.0137) (0.0136)d1990 �0.0029 �0.0048 �0.0020 �0.0035

(0.0133) (0.0132) (0.0131) (0.0131)d1995 �0.0021 �0.0492*** �0.0012 �0.0030

(0.0135) (0.0133) (0.0133) (0.0132)hum65 0.0466*** 19.17*** 0.0478*** 0.0496***

(0.0030) (0.0033) (0.0030) (0.0032)infmort �0.002*** �0.0016*** �0.002*** �0.0017***

(0.0003) (0.0003) (0.0003) (0.0003)lifexp 0.0048*** 0.0054** 0.0045*** 0.0051***

(0.0018) (0.0017) (0.0017) (0.0017)cbr �0.5468 19.17*** �1.4363*** 24.78*

(0.6897) (5.2137) (0.7230) (9.9806)(cbr)2 �487.7*** �665*

(189.44) (306.6897)(cbr)3 3384 5067

(1895) (2985.376)oecd(cbr) 3.7724*** �14.82

(0.8013) (15.7202)oecd(cbr)2 488.5

(585.4747)oecd(cbr)3 �4455

(6899.486)N 792 792 792 792R2/R2 adj. 88.34/88.13 88.8/88.57 88.58/88.36 88.84/88.57F-test 420.5*** 384.2*** 401.2*** 322.6***

F-Joint 16.153*** 0.9286F-Joint 3rd Polyn. – 16.049*** – 7.8147***

Heterosked.a 33.30*** 27.01*** 29.20*** 22.99***

*** Denotes the 1% significance levels.** Denotes the 5% significance levels.* Denotes the 10% significance levels.

a Heterosked. is the heteroskedasticity LM-test by Breusch and Pagan (1979). The standard errors are in parentheses and t-statistics are available uponrequest. We drop the time dummy for 2000. Instead of life expectancy we have used also crude death rate. This variable has negative sign (in contrast to thesign of life expectancy), is statistically significant and the results for the birth rate (cbr) remain the same.


with ei,t being an iid error term. The equation we estimate in the semi-parametric framework is:

PleaseJourna

HUMi;t ¼ b0 þ b1ðdOECDiÞ þ b2ðdRegioniÞ þ b3dt þ /ðni;tÞ þ ei;t :

In the above equation, /(ni,t) is used to capture possible nonlinearities in the relationship between birth rate and humancapital accumulation (all the other variables enter linearly in our specification). We also augment the model to include addi-tional available explanatory variables to guard against omitted variable bias and we further test within the linear frameworkfor possible endogeneity of n, using a Durbin–Wu–Hausman test. Having established the presence of a non-monotonic (non-linear) relationship, we will try to best approximate it by different polynomials both in the model with and without the con-trol variables, selecting the one with the highest explanatory power. Subsequently, we use the specification with the extracontrol variables for conducting sensitivity analysis. The following equation is the one used for checking the robustness ofthe linear specification estimates, and a similar equation is used when we include the optimal polynomial of birth rate:

HUMi;t ¼ b0 þ b1ðdOECDiÞ þ b2ðdRegioniÞ þ b3dt þ b4ni;t þ b5Xi;t þ ei;t



0.050.040.030.020.01

-0.1

5-0

.10

-0.0

50.

000.

05

birth rate

tota

l edu

catio

n

Fig. 1. Semi-parametric plot for the contemporaneous birth rate in the whole sample.


where Xi,t = (lifexp,hum60, infmort) is the vector of control variables. Finally, the equation we estimate by considering all thevariables non-parametrically is:

30 For31 As

polynomconfirmtotal fe

32 The33 The

estimat34 The

compar35 We

correct

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HUMi;t ¼ f ðb0;dOECDi;dRegioni; dt; ni;tÞ þ ei;t :

The fully non-parametric method that is used in this paper was proposed first by Racine and Li (2004) and is appropriatefor mixed data (discrete and continuous variables). We also use the Hsiao et al. (2007) test to check whether the linear modelis well specified against semi/non-parametric alternatives. Finally, we use a significance test30 for the non-parametric regres-sion model in order to ensure that the birth rate is significant in the non-parametric framework.

4.1. Results

A first glance at Table 1 reported above reveals that the birth rate (cbr) is statistically significant and appears in the regres-sion with a negative sign.31 This is consistent with the benchmark theoretical model (see Eq. (5)). The dummies both for LatinAmerica and Sub-Saharan Africa countries are statistically significant and positive in all the specifications, except when humancapital is measured at the secondary-higher level. In this case the region dummies have negative sign. This is an indication thatin these groups of countries the dilution-effect is stronger at the secondary-higher level of education.32 We proceed to estimate asemi-parametric and a fully non-parametric version of the above benchmark model. Clearly, the semi-parametric and the fullynon-parametric specifications33 increase the explanatory power of the model (both have a very similar explanatory power). Fur-thermore, the Hsiao et al. (2007)34 test strongly rejects the hypothesis of a linear specification against the semi-parametric andnon-parametric alternatives – see the p-values of P(Specific) in Table 1 (this paper) and Table B.4 in Boikos et al. (2012). Weproceed in Table 2 to use polynomial terms for the birth rate and also a dummy-variable for OECD countries which interactswith the birth rate in order to control for any heterogeneity between OECD and non-OECD group of countries. Clearly, usingpolynomial terms increases the explanatory power of the model. We only present here results concerning the inclusion of athird degree polynomial, since higher-order polynomials do not appear to be statistically significant. In column B of Table 2the polynomial terms are statistically significant, both individually and jointly (see the F-Joint test). In column C of the sameTable we add an extra variable which captures the interaction between the OECD-dummy and the birth rate. This term is sta-tistically significant and positive. Since the negative coefficient on the birth rate is larger than the positive coefficient on theinteraction term, we infer that the dilution-effect (i.e., the negative impact of the birth rate on human capital investment atthe individual level) in OECD countries is smaller than in non-OECD countries. Finally, in column D we consider the interactionbetween the OECD-dummy and the higher-power terms of the birth rate. By looking at the correspondent t-statistics and the F-Joint test, all the interaction terms are statistically insignificant both individually and jointly, which means that the birth ratebehaves similarly across the different groups of countries.35 We further proceed to test for possible endogeneity of the birth ratein the extended linear model with the extra control variables at our disposal (lifexp,hum60,infmort). The results are presented

this test see Racine (1997).we are going to mention in a moment, the birth rate (cbr) appears with negative sign in the linear specification and there exist nonlinearities (a

ial of the 3rd power). Furthermore, we have the same kind of nonlinearities also when we use total fertility (instead of the crude birth rate), whichs our hypothesis of a nonlinear effect of newborns on per capita human capital accumulation in per-capita terms. We do not present the results with

rtility here, but they are available upon request.results for primary and tertiary level of education are not presented but are available upon request.optimal bandwidths for these methods are found by cross validation and are available upon request. We use the Gaussian kernel and the method of

ion relies on a constant kernel approach.Hsiao test results are showed in the row P(Specific.). In column A, there is the comparison between OLS and fully-non parametric and in column B the

ison is between OLS and semi parametric.have also done separate regressions for OECD and non-OECD countries and still the same third power polynomial of the birth rate appears to be thespecification. These results are available upon request.



0.050.040.030.020.01

-0.1

0-0

.05

0.00

0.05

birth rate

tota

l edu

catio

n

Fig. 2. Semi-parametric plot for the lagged birth rate in the whole sample.


in Table 3. In columns C and D we have similar results as in Table 2 (without the control variables). Again the interaction termsof the polynomial with the OECD dummy is not statistically significant, both individually and jointly, and also the third powerpolynomial remains statistically significant jointly and the signs are the same as in Table 2. Finally, the effect of the birth rate isstill non-monotonic. This result appears to be robust and is the same for OECD as well as for non-OECD countries. Since thespecification with the control variables displays by far the best fit we use it as the basis to conduct the test of endogeneity.We find strong evidence against the null hypothesis of no endogeneity, a result that contradicts the main premise of the bench-mark model that was based on an exogenous birth rate.36 Note that as instruments we use the lagged values of the birth rate.37

We also estimated the semi-parametric model using the lagged birth rate variable in place of the current birth rate in the semi-parametric specification, see the semi-parametric graphs (Figs. 1 and 2). Fig. 1 is the semi-parametric graph for the contempo-raneous value of birth rate and Fig. 2 for the lagged birth rate. Interestingly, both graphs are quite similar, suggesting that thenature of the non-monotonic nonlinear relationship is preserved for different measures of the birth rate, its current and laggedforms.38 A closer look at either figure reveals that a low birth rate has a small positive contribution if any at all but after a givenpoint the dilution effect appears to become strong. This is in line both with theories that support birth control in overpopulatedcountries and also with those that claim that rich countries need to do the opposite.

4.2. Sensitivity analysis

In order to check for the robustness of our results we follow two different ways. First, we use for the initial sample (792observations) the extra control variables mentioned earlier (i.e., life expectancy, infant mortality and human capital at 1965)to test for the significance of the polynomial terms (compare Tables 2 and 3). Secondly, we exclude the OPEC countries asincome earned from oil production potentially distorts the fertility–human capital nexus and we also exclude the Ex-socialistcountries (Eastern Europe and Cuba), since for these countries the existence of a centrally-planned regime might have influ-enced more effectively the dynamics of the birth rate in the past. Results appear in Table B.4 of Boikos et al. (2012).

Table 3 allows us to draw some additional conclusions. First of all, from column B, we observe that the polynomial for thebirth rate preserves the same order and its terms are jointly statistically significant. Furthermore, in columns C and D wehave similar results as in Table 2 (without the control variables). Finally, the effect of the birth rate still appears non-mono-tonic. This result seems to be robust and the same for OECD as well as for non-OECD countries.

In Table B.4 of Boikos et al. (2012) it appears that some of the time-dummies are statistically significant. All the otherresults are qualitatively the same as in Table 1. In particular, we observe that the birth rate is statistically significant andstill has a negative sign. Yet, as before the Hsiao et al. (2007) test strongly rejects the linear specification of the benchmarkmodel. However, the current non-monotonic effect appears stronger (see Fig. 3, Appendix B, in Boikos et al. (2012)). This is inline with our conjecture that the excluded countries have been able to control more effectively their own birth rates, hencemitigating the presence of non-monotonic nonlinear effects.

We can summarize the findings of this section as follows. Superficially, at first there seems to be a negative relationshipbetween the birth rate and human capital by using a linear specification corresponding to the benchmark model developedin Section 2. The benchmark specification is strongly rejected when subjected to rigorous testing and re-estimation of thehuman capital–birth rate nexus using semi-parametric and non-parametric methods reveals the presence of a strongnon-monotonic relationship. However, even in the presence of non-monotonicity the dilution effect dominates anypositive effect that comes from the accumulation effect for most of the observations in our sample. The linear benchmark

36 The Prob > chi2 value of Durbin–Wu–Hausman test for endogeneity is 0.0476 which implies that the null hypothesis for the absence of correlation betweenbirth rate ‘‘n’’ and the error term is rejected.

37 By using the lagged values of the birth rate, the sample size reduces to 792 observations. We label hum 65 the initial stock of human capital at 1965.38 We have also tried additional lagged values of the birth rate and the results remained qualitatively the same.




specification also suffers from endogeneity bias, as the birth rate cannot be taken as exogenous. Adding extra control vari-ables does not change the overall non-monotonic nonlinear pattern of the relationship. The latter result invalidates the mainpremise of the benchmark model that the relationship between birth rate and per-capita human capital investment wouldbe expected to be linear and monotonic.

5. Concluding remarks

This paper has reassessed the long run correlation between demographic change and economic growth from an empirical,as well as a theoretical point of view. In order to accomplish this task, we focused on the fertility rate as our demographicvariable of interest (the birth rate is, indeed, one of the most important demographic variables, as it affects directly popu-lation growth), and on human capital accumulation as the fundamental driver of sustained long run economic growth.While, from the empirical point of view, our motivation was to search for possible non-monotonic effects of the birth rateon human capital investment at the individual level, from the theoretical point of view our main objective was to analyzeand characterize two completely different settings. The first is a simple model in which the birth rate is exogenous and af-fects monotonically and negatively per capita human capital accumulation (linear and negative dilution effect); the second,instead, is a model in which the birth rate is endogenous and the dilution effect can be either linear or nonlinear, and ispotentially non-monotonic. Since countries experience different birth rates and have different economic performances,our major conjecture in building the second model was that the non-monotonic total impact of fertility on economic growthis indeed the consequence of its non-monotonic effect on per capita human capital investment.

The benchmark model is strongly rejected by the data, according to which the birth rate is endogenous and, more impor-tantly, there exist a strong non-monotonicity in the relationship between fertility rate and per capita human capital invest-ment. Because of this evident non-monotonicity, our analysis can ultimately suggest, for any different value of n (and, hence,for any different country in the sample), the direction of the differential total effect of this demographic variable on economicgrowth.

As a whole, our analysis implies that at low levels of the birth rate higher birth rates can surely have a positive growth-effect and that the main policy initiatives here should concentrate on easing the opportunity cost of having children. On theother hand, at high levels of the birth rate, a reduction in these rates can definitely contribute to boost per capita humancapital accumulation and economic growth.

This paper can be extended along different directions, three of which are listed here. The first would be to consider a hu-man capital production technology different from the Lucas’ (1988), and more similar to the Mincer’s (1974) specification(see Bils and Klenow, 2000, Eq. (3), p. 1162). Secondly, in this article we treated human capital as the sole reproducible factor,entering the aggregate production function as the only input. Though, it is well known that economic agents (individuals andfirms) do accumulate and produce by way of a variety of other factor-inputs (think of physical and technological capital, justto mention some notable examples). Building and testing a theory of endogenous fertility in which there is more than onereproducible factor-input and where there are non-monotonicities of the birth rate in the law of motion of one or all of thesefactor-inputs still remains at the top of future research agenda and, thus, represents a further possible extension of the pres-ent paper. Finally, there is also scope to develop a micro-founded model in which the non-monotonic effect of the birth rateon human capital accumulation decisions is derived from individual (rather than aggregate) behavior.

Acknowledgements

We would like to thank an anonymous referee and especially one of the editors (Theodore Palivos) for their useful com-ments. We are also grateful to seminar participants at the universities of Ancona, Barcelona, Milan (DEGIT XVII), Florence andSaint Petersburg (Center for Market Studies and Spatial Economics) for insightful comments and remarks on an earlier versionof this paper. In particular, we wish to thank Jacub Growiec, Luca Guerrini, Xavier Raurich and Jacques Thisse for discussionand constructive suggestions.

Appendix. Sample of countries for which we use data

OECD Countries: Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Finland, France, Germany, Greece,Hungary, Iceland, Ireland, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Republic of Korea, Spain,Sweden, Switzerland, Turkey, UK, and US.

Non-OECD Countries: Afghanistan, Algeria, Argentina, Bahrain, Bangladesh, Barbados, Bolivia, Botswana, Brazil, Bulgaria,Cameroon, Central African Republic, Colombia, Costa Rica, Cuba, Cyprus, Dominican Republic, Ecuador, El Salvador, Fiji, Gha-na, Guatemala, Guyana, Haiti, Honduras, Hong Kong, India, Indonesia, Iran (Islamic Republic of), Iraq, Israel, Jamaica, Jordan,Kenya, Kuwait, Lesotho, Liberia, Malawi, Malaysia, Mali, Mauritius, Mozambique, Myanmar, Nepal, Nicaragua, Niger, Paki-stan, Panama, Papua New Guinea, Paraguay, Peru, Philippines, Romania, Senegal, Sierra Leone, Singapore, South Africa, SriLanka, Sudan, Swaziland, Syrian Arab Republic, Thailand, Togo, Trinidad and Tobago, Tunisia, Uganda, Uruguay, Venezuela,Zambia, and Zimbabwe.




OPEC Countries: Algeria, Ecuador, Indonesia, Iran (Islamic Republic of), Iraq, Kuwait, and Venezuela.Eastern European/Ex-Socialist Countries: Bulgaria, Czech Republic, Cuba, Hungary, Poland, and Romania.Latin America Countries: Argentina, Barbados, Bolivia, Brazil, Chile, Colombia, Costa Rica, Cuba, Dominican Republic,

Ecuador, El Salvador, Guatemala, Guyana, Haiti, Honduras, Jamaica, Mexico, Nicaragua, Panama, Paraguay, Peru, Trinidadand Tobago, Uruguay, and Venezuela.

Sub-Saharan Africa Countries: Botswana, Cameroon, Central African Republic, Ghana, Kenya, Lesotho, Liberia, Malawi,Mali, Mauritius, Mozambique, Niger, Senegal, Sierra Leone, South Africa, Sudan, Swaziland, Togo, Uganda, Zambia, andZimbabwe.

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